System and method for controlling superconducting qubits using single flux quantum logic

ABSTRACT

A system and method for controlling superconducting qubits is provided. In some aspects the method includes assembling, using a controller of a quantum computing system, a pulse subsequence that comprises pairs of voltage pulses timed symmetrically with respect to a period corresponding to a qubit frequency of a superconducting qubit in the quantum computing system. The method also includes generating, using the controller, a pulse sequence using a repetition of a pulse subsequence. The method further includes controlling the superconducting qubit by applying the pulse sequence to the superconducting qubit using a single flux quantum (“SFQ”) driver coupled thereto.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

This invention was made with government support under 1720304 awarded bythe National Science Foundation. The government has certain rights inthe invention.

BACKGROUND

The field of the disclosure is related to superconducting circuits. Moreparticularly, the disclosure relates to systems and methods forcontrolling superconducting qubits using single flux quantum (“SFQ”)circuits.

In the field of quantum computation, the performance of quantum bits(“qubits”) has advanced rapidly in recent years, with severalpreliminary multi-qubit initiatives aiming toward surface codearchitectures. In contrast to classical computational methods that relyon binary data stored in the form of definite on/off states, orclassical bits, qubits take advantage of the quantum mechanical natureof quantum systems to store and manipulate data. Specifically, quantumsystems can be described by multiple quantized energy levels or states,and can be represented probabilistically using a superposition of thosestates.

Among several implementations currently being pursued,superconductor-based qubits present good candidates for quantumcomputation. This is because superconducting materials have inherentlylow dissipation that, in principle, can produce coherence timesnecessary for performing useful calculations. For instance, qubits basedon Josephson tunnel junctions, which include two superconductingelectrodes separated by a thin insulator, are advantageous due to theirstrongly nonlinear behavior. Specifically, Josephson-based devices allowfor breaking the degeneracy between different transition frequencies,and thereby restrict system dynamics to specific quantum states. Inaddition, complex superconducting circuits can be micro-fabricated usingconventional integrated-circuit processing techniques. This allowsscaling to architectures that include a large number of qubits.

A fault-tolerant scalable quantum computer can provide a computationalpower far exceeding that of a classical computer, and superconductingqubits are a promising way to build such a machine. However, large-scalequantum information processing based on surface codes imposes strictchallenges on qubit operation and control. For instance, by someestimates, a general-purpose fault-tolerant quantum computer will likelyinclude millions of physical qubits. Using current implementations,controlling such large-scale quantum computer through qubitmanipulation, error detection, and readout, would involve a massivehardware overhead.

Conventionally, qubits are controlled using pulses generated bysingle-sideband modulation of a microwave carrier tone. Accurate controlof both the in-phase and quadrature pulse amplitudes allows arbitraryrotations on the Bloch sphere. However, utilizing microwave pulsesintroduce the possibility of crosstalk between neighboring qubitchannels of a qubit array. To minimize crosstalk, different qubits inthe array are often biased at different operating frequencies. Thisapproach also makes it possible to address a large-scale multi-qubitarray with a relatively small number of carrier tones, which results insignificant hardware savings.

In some approaches, control waveforms are recycled and used across aqubit array. However, it is not clear that recycling waveforms allowshigh-fidelity control. This is because such waveforms represent theconvolution of the applied waveforms and transfer functions of thewiring in the cryostat system. However, transfer functions are generallynot well controlled, and can vary substantially across the array.Moreover, having separate high-bandwidth control lines for each qubitchannel entails a massive heat load on the milli-Kelvin stage of thecryostat system. Furthermore, the significant latency associated withthe round trip signal travel from the quantum array to theroom-temperature classical coprocessor will limit the performance of anyscheme used for high-fidelity projective measurement and feedback tostabilize the qubits in the array.

Given the above, there exists a need for systems and methods yieldingscalable quantum computation that includes the ability to perform rapidhigh-fidelity control and measurement of both single qubits andmulti-qubit parity, while controlling the resources utilized.

SUMMARY

The present disclosure overcomes the drawbacks of previous technologies.In one aspect of the disclosure, a quantum computing system is provided.The system includes a qubit architecture comprising a superconductingqubit described by a qubit frequency, and a single flux quantum (“SFQ”)driver coupled to the superconducting qubit, wherein the SFQ driver isconfigured to provide a pulse sequence to control the superconductingqubit, the pulse sequence being generated using a repetition of a pulsesubsequence that comprises pairs of voltage pulses timed symmetricallywith respect to a period corresponding to the qubit frequency.

In another aspect of the disclosure, a method for controllingsuperconducting qubits is provided. The method includes assembling,using a controller of a quantum computing system, a pulse subsequencethat comprises pairs of voltage pulses timed symmetrically with respectto a period corresponding to a qubit frequency of a superconductingqubit in the quantum computing system. The method also includesgenerating, using the controller, a pulse sequence using a repetition ofa pulse subsequence. The method further includes controlling thesuperconducting qubit by applying the pulse sequence to thesuperconducting qubit using a single flux quantum (“SFQ”) driver coupledthereto.

The foregoing and other aspects and advantages of the invention willappear from the following description. In the description, reference ismade to the accompanying drawings which form a part hereof, and in whichthere is shown by way of illustration a preferred embodiment of theinvention. Such embodiment does not necessarily represent the full scopeof the invention, however, and reference is made therefore to the claimsand herein for interpreting the scope of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawing(s) will be provided by the Office upon request and paymentof the necessary fee.

FIG. 1A is a schematic of a quantum computing system, in accordance withthe present disclosure.

FIG. 1B is a schematic of example superconducting quantum processor(s)for the quantum computing system of FIG. 1A.

FIG. 1C is circuit diagram of an example superconducting quantumprocessor in which a SFQ driver is capacitively coupled to a transmonqubit, in accordance with the present disclosure.

FIG. 2A is an illustration showing a resonant pulse sequence and thequbit trajectory traversed on the front hemisphere (i.e <x><0) of theBloch sphere using the resonant pulse sequence.

FIG. 2B is an illustration showing a symmetric pulse sequence, inaccordance with aspects of the present disclosure, and the qubittrajectory traversed on the front hemisphere (i.e <x> <0) of the Blochsphere using the symmetric pulse sequence.

FIG. 3 is a map showing qubit frequencies permitting high-fidelitycontrol, in accordance with the present disclosure.

FIG. 4 is an illustration showing trajectories on the Bloch sphere forqubits initialized in states |x₊

(green), |y₊

(purple), and |z₊

(red).

FIG. 5 is a graphical illustration showing a subsequence optimizationalgorithm, in accordance with aspects of the present disclosure.

FIG. 6 is a graphical illustration showing gate infidelity versus tipangle δθ.

FIG. 7 is a graphical illustration showing SCALLOP subsequences, inaccordance with aspects of the present disclosure.

FIG. 8 are graphs showing leakage into non-computational states for thesequence of FIG. 6 corresponding to a 4.89201 GHz qubit.

FIG. 9 are graphs showing sensitivity of gate fidelity to variation inqubit frequency and anharmonicity.

FIG. 10 is a flowchart setting forth steps of a process, in accordancewith aspects of the present disclosure.

DETAILED DESCRIPTION

Conventional methods for controlling a qubit generally utilize resonantmicrowave pulses to realize arbitrary rotations over the Bloch sphere,which is a geometrical representation of the state of a two-levelsystem. Amplitude modulation of the microwave concentrates drive powerat the frequency of interest, and the pulse shape minimizes power atnearby transition frequencies to avoid excitation out of the qubitmanifold. However, such approaches are limited in their applicability toscalable quantum computing systems due to the complex and expensiveresources required.

Computational burden may be reduced by integrating a classicalcoprocessor unit with the qubits at the milli-Kelvin stage. In thisapproach, coherent control may be achieved by irradiating qubits withtrains of quantized flux pulses produced using Single Flux Quantum (SFQ)digital logic. Generally, SFQ digital logic circuits generate,manipulate and store classical bits of information, or logical “0” and“1” values, using voltage pulses, or fluxons, that propagateballistically along passive superconducting microstrip lines or activeJosephson transmission lines. That is, classical bits of information arestored by way of a presence or absence of a phase slip across aJosephson junction in a given clock cycle. The phase slip results in avoltage pulse whose time integral is precisely quantized to thesuperconducting flux quantum Φ₀=h/2e. For typical parameters, SFQ pulseamplitudes are of order 1 mV and pulse durations are around 2 ps, whichis roughly two orders of magnitude shorter than the typical qubitoscillation period. As a result, each SFQ pulse imparts a deltafunction-like kick to the qubit that induces a coherent rotation in thequbit subspace.

The fidelity achieved using such SFQ-generated pulse trains, however,may be limited by leakage out of the computational subspace. Forexample, using a resonant sequence where SFQ pulse spacing is matched tothe qubit oscillation period, fidelities around 99.9% may be obtainedfor gate times around 20 ns and typical values of qubit anharmonicity.These fidelities are likely insufficient for fault-tolerant operationsin a large-scale surface code array. To achieve SFQ-based gates withhigher fidelity, SFQ bits can be clocked to the qubit at a higher rate.In one previous proof-of-principle demonstration, a genetic algorithmwas used to derive SFQ pulse trains with variable pulse-to-pulse timingsthat showed low leakage and gate fidelities better than 99.99%. However,a large number of bits was required to achieve the higher fidelity,which presents serious obstacles to achieving practical large-scaleimplementations. In addition, that approach provided no intuition as towhy a particular sequence would yield good performance.

By contrast, the present disclosure introduces a novel method to derivehardware-efficient SFQ control sequences for scalable qubit control,referred to herein as SCALable Leakage Optimized Pulse Sequences(SCALLOPS). As detailed below, qubit control sequences may beconstructed using short subsequences of classical bits that can berepeatedly streamed. Leakage is minimized at the subsequence level, andbecause the subsequences are short, it is possible to perform efficientsearch over the subsequence space in order to optimize gate fidelity.For SFQ clock frequency a factor of a few greater than thecharacteristic qubit frequency, high-fidelity qubit rotations can beachieved for a large number of discrete qubit frequencies, as requiredfor low-crosstalk control of a large-scale qubit array designed toimplement the surface code.

Turning now to FIG. 1A, an example system 100 for use in quantuminformation processing or quantum computation, in accordance with thepresent disclosure, is shown. In some embodiments, the system 100 mayinclude a controller 102 and signal input/output (I/O) hardware 104 incommunication with the controller 102. The system 100 may also includeone or more superconducting processors 106 contained in a housing unit108, where the superconducting processor(s) 106 is configured to performa variety of quantum computations or quantum information processing. Inaddition, the system 100 may also include various interface hardware 110for communicating and controlling signals between the signal I/Ohardware 104 and the superconducting processor(s) 106.

The signal I/O hardware 104 may include various electronic systems,hardware and circuitry capable of a wide range of functionality. Forexample, the signal I/O hardware 104 may include various voltagesources, current sources, signal generators, amplifiers, filters,digitizers, mixers, multiplexers, voltmeters, digital/analogoscilloscopes, data acquisition cards, digital/analog signal controllersand/or processors, modulators, demodulators, logic blocks, and otherequipment. In some implementations, the signal I/O hardware 104 mayinclude a field programmable gate array (FPGA) configured to generateand provide multi-bit patterns to be used by an SFQ driver to controlone or more qubits.

In general, the controller 102 may direct the signal I/O hardware 104 toprovide various signals to the superconducting processor(s) 106, as wellas detect signals therefrom via the interface hardware 110. In someimplementations, the controller 102 may also control various otherequipment of the system 100, such as various pumps, valves, and soforth. In some aspects, the controller 102 may include a programmableprocessor or combination of processors, such as central processing units(CPUs), graphics processing units (GPUs), and the like. As such, thecontroller 102 may be configured to execute instructions stored in anon-transitory computer readable-media. In this regard, the controller102 may be any computer, workstation, laptop or other general purpose orcomputing device. Additionally, or alternatively, the controller 102 mayalso include one or more dedicated processing units or modules that maybe configured (e.g. hardwired, or pre-programmed) to carry out steps, inaccordance with aspects of the present disclosure.

The housing unit 108 is configured to control the environment to whichthe superconducting processor(s) 106 is exposed. For instance, thehousing unit 108 may include various components and hardware configuredto control the temperature of the superconducting processor(s) 106, aswell as the liquid and/or gas mixture surrounding the superconductingprocessor(s) 106. In addition, the housing unit 108 may also beconfigured to control external noise signals, such as strayelectromagnetic signals. To this end, the housing unit 108 may includevarious shielding units and filters. By way of example, the housing unit108 may include, or be part of, a dilution refrigerator, or otherlow-temperature system or cryostat, that is capable of operating over abroad range of temperatures, including temperatures less than thecritical temperature of the superconductor materials in thesuperconducting processor(s) 106 (e.g. temperatures less than 4 Kelvin).

The interface hardware 110 provides a coupling between the signal I/Ohardware 104 and the superconducting quantum processor(s) 106, and mayinclude a variety of hardware and components, such as various cables,wiring, RF elements, optical fibers, heat exchanges, filters,amplifiers, stages, and so forth.

As shown in FIG. 1A, the superconducting processor(s) 106 may include aqubit architecture 112 connected to control circuitry 114 by way ofvarious control coupling(s) 116. The qubit architecture 112 may includeany number of qubits configured in any manner. In some implementations,the qubit architecture 112 may include one or more transmon qubits (e.g.xmon qubits). However, the qubit architecture 112 may include otherqubit types including charge qubits, flux qubits, phase qubits, andothers.

The control circuitry 114 may be in communication with the signal I/Ohardware 104, and configured to control qubits in the qubit architecture112 by providing various control signals thereto. In someimplementations, as shown in FIG. 1B, the control circuitry 114 includesan SFQ driver 120 that is coupled to the qubit architecture 112. Forpurposes of illustration, FIG. 1C shows an example diagram of an SFQdriver 120 capacitively coupled to a transmon qubit 124. In general, theSFQ driver 120 may be configured to generate and provide a pulsesequence to control qubits in the qubit architecture 112. This may beaccomplished using by way of the signal I/O hardware 104, which asdirected by the controller 102, may initiate and control the timing,intensity and repetition of voltage pulses provided by the SFQ driver120.

In accordance with aspects of the present disclosure, the pulse sequencegenerated by the SFQ driver 120 includes a plurality of voltage pulses(i.e. SFQ pulses) having variable pulse-to-pulse temporal intervals. Aswill be described in detail below, the length of the voltage pulsesequence (i.e. the number of pulses in each sequence) and thepulse-to-pulse temporal spacing, or timing intervals between the pulses,may be optimally selected to perform effective control (e.g.high-fidelity operations), minimize leakage outside the computationalspace and reduce hardware overhead.

Other example control signals directed by the control circuitry 114 tothe qubit architecture 112 may also include microwave irradiationsignals, current signals, voltage signals, magnetic signals, and so on.To this end, the control circuitry 114 may include various othercircuitry, including any number of linear and non-linear circuitelements, such as Josephson junctions, inductors, capacitors, resistiveelements, superconductive elements, transmission lines, waveguides,gates, and the like.

The control couplings 116 providing a communication between the qubitarchitecture 112 and control circuitry 114 may configured to transmit,modulate, amplify, or filter, the pulse sequence generated using thecontrol circuitry 114. Such control couplings 116 may include variouscircuitry, including capacitive or inductive elements, passivesuperconducting microstrip lines, active Josephson transmission lines,including any number of Josephson junctions, and so forth.

Referring again to FIG. 1A, the qubit architecture 112 may also beconnected to readout circuitry 118 via readout coupling(s) 122. Thereadout circuitry 118 may be configured to perform readout on qubits inthe qubit architecture 112, and provide corresponding signals to thesignal I/O hardware 104. As non-limiting examples, the readout circuitry118 may include various resonant cavities, logic circuits, as well asany number of linear and non-linear circuit elements, such as Josephsonjunctions, inductors, capacitors, resistive elements, superconductiveelements, transmission lines, waveguides, gates, and the like. In someaspects, the controller 102 may direct the signal I/O hardware 104 toprovide signals for modulating or tuning the control couplings 116and/or readout couplings 122.

In certain desired configurations, the control couplings 116 and/orreadout couplings 122 may be designed such that non-equilibriumquasiparticles generated in the control circuitry 114 or readoutcircuitry 118 are isolated from the qubit architecture 112 in a mannerintended to avoid the introduction of degrees of freedom leading toquantum decoherence. For example, quasiparticle poisoning can bemitigated by avoiding direct galvanic connection between the signal andground traces of the qubit architecture 112 and the control circuitry114 and/readout circuitry 118. This may be achieved using a modularapproach, as described in R. McDermott et al. (“Quantum-ClassicalInterface Based on Single Flux Quantum Digital Logic,” Quantum Sci.Technol. 3, 024004 (2018)), which is incorporated herein by reference,in its entirety.

The present SCALLOPS approach will now be described in more detail. Byway of illustration, a fixed SFQ clock frequency of 25 GHz isconsidered, so that SFQ pulses are delivered to the qubit at intervalsthat are integer multiples of the 40 ps clock period. In addition,transmon qubits with fixed anharmonicity (ω₁₀-ω₂₁)/2π of 250 MHz areconsidered, where ω₁₀≡ω_(q) is the qubit transition frequency and ω₂₁ isthe transition frequency between the qubit |1

state and the noncomputational |2

state. Finally, for the sake of concreteness high fidelity for a singlegate, the Y_(π/2) rotation, is targeted. However, it may be appreciatedthat the present SCALLOPS approach is readily applied to varioussingle-qubit operations, as well as various other qubit types includingcharge qubits, flux qubits, phase qubits, and others. In addition, asdetailed below, high-fidelity control is possible for up to and over 20qubit frequencies spanning the range from 4.5 to 5.5 GHz, with optimalgate fidelity achieved when an integer multiple of the qubit frequencyis matched to an integer multiple of the clock frequency.

To start, a conventional transmon qubit coupled via a small capacitanceC to an SFQ driver is considered, as shown in FIG. 1C. The SFQ drivermay be modeled as a time-dependent voltage source V_(SFQ)(t). First, theHamiltonian of the undriven transmon may be written as:

$\begin{matrix}{{H_{fr} = {\frac{{\hat{Q}}^{2}}{2C^{\prime}} - {E_{J}\cos\hat{\varphi}}}},} & (1)\end{matrix}$where {circumflex over (Q)} and {circumflex over (ϕ)} are the charge andphase operators of the transmon, and E_(J) is the transmon Josephsonenergy, and C′=C_(c)+C is the sum of the coupling capacitance C_(c) andthe transmon self-capacitance C. H_(fr) can be diagonalized in closedform, with the resulting energy eigenfunctions

ϕ|γ

and energies E_(γ) represented by the Mathieu functions andcoefficients.

Interaction between the transmon and the SFQ pulse driver adds thefollowing term to the Hamiltonian:

$\begin{matrix}{{H_{SFQ}(t)} = {\frac{C_{c}}{C^{\prime}}{V_{SFQ}(t)}{\hat{Q}.}}} & (2)\end{matrix}$where the SFQ pulse V_(SFQ)(t) satisfies the condition

∫_(−∞)^(∞)V_(SFQ)(t)dt = Φ₀.Since we pulse width (typically around 2 ps) is much less than theLarmor period (typically around 200 ps) of the transmon, the SFQ pulsemay be modeled by a Dirac delta function: V_(SFQ)(t)=Φ₀δ(t). The chargeoperator can be constructed in the basis of |γ

once H_(fr) is diagonalized. The free evolution of the transmon thenbecomes

$\begin{matrix}{{U_{fr}(t)} = {{\exp( {{- \frac{i}{h}}{\sum{ \gamma \rangle\langle \gamma  E_{\gamma}t}}} )}.}} & (3)\end{matrix}$

The time evolution for a transmon subjected to an SFQ pulse isU _(SFQ)=exp(−iΦ ₀(C _(c) /C′){circumflex over (Q)}).  (4)

In some implementations, it may be advantageous to restrict the size ofthe transmon Hilbert space in order to accelerate the search forhigh-fidelity pulse sequences. Since leakage outside the computationsubspace is dominated by the population of the first non-computationalstate |2

, the transmon can be truncated to a three-level qutrit. However,validation may also be performed using more complete models the transmonthat include up to 7 states. Within the three-level subspace, theoperators H_(fr) and H_(SFQ)(t) take the form

$\begin{matrix}{{H_{fr}^{(3)} = {\frac{{\hslash\omega}_{q}}{2}\hat{\sum\limits_{z}}}},{{H_{SFQ}^{(3)}(t)} = {\frac{{\hslash\omega}_{y}(t)}{2}\hat{\sum\limits_{y}}}},} & (5) \\{{\hat{\sum\limits_{z}}{= \begin{bmatrix}0 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & {4 - {2\eta}}\end{bmatrix}}},{\hat{\sum\limits_{y}}{= {{i\begin{bmatrix}0 & {- 1} & 0 \\1 & 0 & {- \lambda} \\0 & \lambda & 0\end{bmatrix}}.}}}} & (6)\end{matrix}$

Here, ω_(y)(t)=−(2V(t)/h)((C_(c)/C′)

1|{circumflex over (Q)}|0

, with η=1−ω₂₁/ω_(q) representing the fractional anharmonicity of thetransmon, and λ=

2|{circumflex over (Q)}|1

/

1|{circumflex over (Q)}|0

.

The three-level matrix form for the free evolution of the transmon andfor the evolution of the transmon subjected to a single SFQ pulse maythen be derived. Specifically, the free evolution is given by thediagonal matrix U_(fr) ⁽³⁾(t)=exp(−iω_(q)t{circumflex over (Σ)}_(z)/2).In the qubit subspace, the effect of U_(fr) ⁽³⁾ is a precession at arate of ω_(q). For the time evolution under a single SFQ pulse, one maywrite

$\begin{matrix}{{U_{SFQ}^{(3)} = {{\exp( {{- \frac{i\hat{\sum\limits_{y}}}{2}}{\int_{- \infty}^{\infty}{{\omega_{y}(t)}{dt}}}} )} = {\exp( \frac{{- i}\;{\delta\theta}\hat{\sum\limits_{y}}}{2} )}}},} & (7)\end{matrix}$

where

$\begin{matrix}{{\delta\theta} = {{\int_{- \infty}^{\infty}{{\omega_{y}(t)}{dt}}} = {( {2{\Phi_{0}/\hslash}} )( {C_{c}/C^{\prime}} )\langle {1{\overset{\hat{}}{Q}}0} \rangle}}} & (8)\end{matrix}$

is the tip angle associated with the single SFQ pulse. Using theCayley-Hamilton theorem on {circumflex over (Σ)}_(y), the property

$\hat{\overset{3}{\sum\limits_{y}}}{= {( {\lambda^{2} + 1} )\hat{\sum\limits_{y}}}}$may be obtained, which yields

$\hat{\overset{n}{\sum\limits_{y}}}{= {( {\lambda^{2} + 1} )^{\frac{n - 2}{2}}\hat{\overset{2}{\sum\limits_{y}}}}}$for even n and

$\hat{\sum\limits_{y}^{n}}{= {( {\lambda^{2} + 1} )^{\frac{n - 1}{2}}\hat{\sum\limits_{y}}}}$for odd n. Expanding and regrouping Eqn. 7 then gives

$\begin{matrix}{U_{SFQ}^{(3)} = {\hat{1} + {\sum\limits_{{even},{n \geq 2}}^{\infty}{\frac{( {\lambda^{2} + 1} )^{\frac{n - 2}{2}}( {{- i}\;{{\delta\theta}/2}} )^{n}}{n!}{\hat{\sum\limits_{y}^{2}}{{+ {\quad\quad}}{\quad{\underset{{odd},{n \geq 1}}{\overset{\infty}{\quad\sum}}\frac{( {\lambda^{2} + 1} )^{\frac{n - 1}{2}}( {{- i}\;{{\delta\theta}/2}} )^{n}}{n!}\hat{\sum\limits_{y}.}}}}}}}}} & (9)\end{matrix}$

The two sums in Eqn. 9 yield

$\begin{matrix}{{U_{SFQ}^{(3)} = {\frac{1}{\kappa^{2}} \times \begin{bmatrix}{\lambda^{2} + {\cos( {{\kappa\delta\theta}/2} )}} & {- {{\kappa sin}( {{\kappa\delta\theta}/2} )}} & {2{{\lambda sin}^{2}( {{\kappa\delta\theta}/4} )}} \\{{\kappa sin}( {{\kappa\delta\theta}/2} )} & {\kappa^{2}{\cos( {{\kappa\delta\theta}/2} )}} & {- {{\kappa\lambda sin}( {{\kappa\delta\theta}/2} )}} \\{2{{\lambda sin}^{2}( {{\kappa\delta\theta}/4} )}} & {{\kappa\lambda sin}( {{\kappa\delta\theta}/2} )} & {1 + {\lambda^{2}{\cos( {{\kappa\delta\theta}/2} )}}}\end{bmatrix}}},} & (10)\end{matrix}$

where κ=√λ²+1.

To see the effect of a single SFQ pulse, the time evolution is comparedto a y-rotation by angle δθ in the qubit subspace:

$\begin{matrix}{Y_{\delta\theta} = {\begin{bmatrix}{\cos( {{\delta\theta}/2} )} & {- {\sin( {{\delta\theta}/2} )}} \\{\sin( {{\delta\theta}/2} )} & {\cos( {{\delta\theta}/2} )}\end{bmatrix}.}} & (11)\end{matrix}$

It may then be observed that: (1) within the three-level model, the SFQpulse provides a rotation in the qubit subspace that is slightly smallerthan δθ; and (2) leakage from state |1

to state |2

is first order in δθ, while leakage from state |0

to state |2

is second order in δθ.

The above analysis shows that coherent qubit rotations with a single SFQpulse incurs significant excitations of non-computational states. Bycontrast, and in accordance with the present disclosure, compositesequences consisting of multiple SFQ pulses spaced in time byappropriately selected intervals may be used to achieve low leakage andhigh fidelity. As such, a high-speed SFQ clock delivering a sequence ofpulses to a transmon according to a vector of binary variables S may beconsidered, where S_(i)=0 if no SFQ pulse is applied on the i^(th) clockedge and S_(i)=1 if an SFQ pulse is applied. Using these expressions,the total time evolution operator of the gate U_(G), time ordered interms of clock edges, can be written as:

$\begin{matrix}{U_{G} = {T\{ {{\prod\limits_{i}^{N_{C}}( {\delta_{S_{i}1}{U_{fr}( {{T_{c)}U_{SFQ}} + {\delta_{S_{i}0}{U_{fr}( T_{c} )}}} )}} \}},} }} & (12)\end{matrix}$

where T is the time ordering operator. Here, N_(c) is the number ofclock cycles in the sequence and is the clock period. The fidelity ofthe gate U_(G) may then be evaluated as

$\begin{matrix}{F_{avg} = {\frac{1}{6}{\sum\limits_{{\alpha\rangle} \in v}{{\langle {\alpha{{U_{G}^{\dagger}Y_{\pi/2}}}\alpha} \rangle }^{2}.}}}} & (13)\end{matrix}$

The summation in Eqn. 13 runs over the six states v aligned along thecardinal directions of the Bloch sphere

$\begin{matrix}{{ x_{\pm} \rangle = \frac{ 0 \rangle \pm  1 \rangle}{\sqrt{2}}},{ y_{\pm} \rangle = \frac{ 0 \rangle \pm {i 1 \rangle}}{\sqrt{2}}},{ z_{+} \rangle =  0 \rangle},{ {z\_} \rangle =  1 \rangle},} & (14)\end{matrix}$

and where the Y_(π/2) gate may be represented by the following matrix inthe qubit subspace:

$\begin{matrix}{Y_{\pi/2} = {\frac{1}{\sqrt{2}}{( {{ 0 \rangle\langle 0 } + { 1 \rangle\langle 1 } + { 1 \rangle\langle 0 } - { 0 \rangle\langle 1 }} ).}}} & (15)\end{matrix}$

The crux of the problem then becomes the selection of a proper S so thatU_(G) becomes a high-fidelity Y_(π/2) gate.

In the simplest scheme, as illustrated in FIG. 2A, a period train of SFQpulses synchronized to the qubit oscillation period, T_(q)=2π/ω_(q), maybe applied for coherent control. Because U_(fr)(T_(c))={circumflex over(1)} in the qubit subspace, only the desired y-rotations can be inducedby U_(fr)(T_(c)) and U_(SFQ). However, leakage out of the qubit spacecan be significant.

To attain higher fidelity, the present disclosure envisions employingmore sophisticated sequences, with clock SFQ pulses delivered at higherrates compared to the qubit oscillation period. However, when the qubitis no longer resonant with the SFQ clock, U_(fr)(T_(c))≈{circumflex over(1)} and the time evolution of an arbitrary sequence of U_(fr)(T_(c))and U_(SFQ) is generally not confined to y-rotations. Therefore, toensure high overlap with the target y-rotation, a sequence is assembledusing symmetric pairs of SFQ pulses. As shown in FIG. 2B, the symmetricpairs of SFQ pulses are delivered to the qubit at times ϕ/ω_(q) and(2mπ−2ϕ)/ω_(q), for some integer m, and occur symmetrically with respectto the time mT_(q)/2.

The symmetric pair of pulses can be represented by the tuple notation(m, ϕ). As an example, a resonant sequence may be written in terms ofsymmetric pairs, where the first and last pulses form the pair(N_(q),0), the second and penultimate pulses form the pair (N_(q),2π),and so forth. In general, the sequence can be described as a set ofsymmetric pairs (N_(q),2πk) for each k between 0 and N_(q)/2.

To verify that application of symmetric pulse pairs has the net effectof a y-rotation within the qubit subspace, the time evolution operatorU_((m,ϕ)) associated with symmetric pair (m, ϕ) may be inspected:U _((m,ϕ)) =U _(fr)(ϕ/ω_(q))U _(SFQ) U _(fr)((2mπ−2ϕ)/ω_(q))×U _(SFQ) U_(fr)(ϕ/ω_(q)).  (16)

Using U_(SFQ) from Eqn. 10, and expanding to first order in δθ, oneobtains

$\begin{matrix}{U_{({m,\phi})} = {\begin{bmatrix}1 & {{- {\cos(\phi)}}{\delta\theta}} & 0 \\{\cos(\phi){\delta\theta}} & 1 & {- {\lambda\mu\delta\theta}} \\0 & {\lambda\tau\delta\theta} & \ldots\end{bmatrix} + {\Theta( {\delta\theta}^{2} )}}} & (17)\end{matrix}$

where

$\begin{matrix}{\mu = {{\exp( \frac{{im}\;{\pi( {{2\omega_{q}} + \omega_{21}} )}}{\omega_{q}} )}{{\cos( {\frac{\omega_{21}}{\omega_{q}}( {{m\;\pi} - \phi} )} )}.}}} & (18)\end{matrix}$

To first order in δθ, it may be observed that u_((m,ϕ)) is indeed ay-rotation in the qubit subspace. Moreover, the dependence of leakage onthe timing of the symmetric pair through ϕ provides a degree of freedomthat allows for optimizing subsequences to minimize leakage errors, asfurther detailed below. To note, although it might be tempting to setμ=0 by appropriate selection of ϕ, and thereby eliminate the 1-2transition, the 0-1 transition will become very weak as a side effect.In fact, there is an analogous composite microwave pulse method thatexploits a restricted form of this idea corresponding to m=1. However,the gate performance is no better than that of nave Gaussian pulses.

As further explained below, the construct of symmetric pairs becomesparticularly advantageous when it is extended to the case of multiplepairs (m_(i),ϕ_(i)) applied at times ϕ_(i)/ω_(q) and (2m_(i)π−ϕ_(i)) fori∈N, as shown in FIG. 2B. Noteworthy, although the pulse pairs dointerfere with each other because they generally do not commute, theresulting error is sufficiently small for practical choices of δθ.

In general, the qubit oscillation period will not be commensurate withthe SFQ clock, so that the optimal delivery times of the symmetric pairswill not exactly coincide with SFQ clock edges. As a result, it isnecessary to round a symmetric pair to a particular pair of clock edgesn_(i) and n_(j). To note, n_(i) and n_(j) can preserve the symmetryprecisely if the times at which the pulses are applied are symmetricwith respect to mT_(q)/2 for some integer to:

$\begin{matrix}{{\frac{1}{2}( {{\frac{n_{i}}{N_{c}} \cdot \frac{2\pi\; N_{q}}{\omega_{q}}} + {\frac{n_{j}}{N_{c}} \cdot \frac{2\pi\; N_{q}}{\omega_{q}}}} )} = {{mT}_{q}/2}} & (19)\end{matrix}$

where N_(c) is the number of clock cycles and N_(q) is the number ofqubit cycles in the sequence. This condition is equivalent to theexpression

$\begin{matrix}{{A_{sym} = {{{{( {\frac{n_{i}}{N_{c}}N_{q}} ){mod}\; 1} + {( {\frac{n_{j}}{N_{c}}N_{q}} ){mod}\; 1} - 1}} = 0}},} & (20)\end{matrix}$

where A_(sym) is a measure of the violation of symmetry due to mismatchbetween the SFQ clock and the qubit oscillation period. Empirically itwas found that coherent pulse errors are acceptably small for pulsepairs delivered at times such that A_(sym)<0.05. In the following, pulsepairs that are termed symmetric are understood to satisfy thiscondition.

The delivery of SFQ pulses to the qubit as symmetric pairs constrainsthe time evolution to the desired y-rotation. However, it is not obvioushow to control multiple qubits resonating at different frequencies, asrequired by the surface code. For a qubit frequency that is not asubharmonic of the SFQ clock frequency, the concern is that mismatchbetween the qubit oscillation period and the SFQ clock will lead tophase errors, as the precession of the qubit during the gate is not aninteger number of qubit cycles.

To avoid such phase errors, the key is to tune the qubit frequency suchthat the total gate time T_(g) corresponds to both an integer numberN_(c) of clock cycles T_(c) and an integer number N_(q) of qubit cyclesT_(q), so that T_(g)=N_(c)T_(c)=N_(q)T_(q). This relation translatesinto the following frequency matching condition:

$\begin{matrix}{\frac{N_{q}}{\omega_{q}} = \frac{N_{c}}{\omega_{c}}} & (21)\end{matrix}$

where ω_(c)=2πf_(c) is the angular frequency of the clock. To findfrequencies that satisfy Eqn. 21, a map, as illustrated in FIG. 3, maybe generated and used. Specifically, each grid point in FIG. 3represents a qubit oscillation frequency that satisfies Eqn. 21, wherethe grid points highlighted in color span qubit frequencies from about4.5 to 5.5 GHz, assuming an SFQ clock frequency of about 25 GHz. Ofcourse, it may readily be understood that such map may vary depending onthe selected SFQ clock frequency. For a small range of frequenciesaround qubit operating points satisfying Eqn. 21, accurate qubit controlis possible.

From the frequency-matching relation Eqn. 21, it is clear that longergate times will permit high-fidelity control of a larger number ofdistinct qubit frequencies. However, in accordance with aspects of thepresent disclosure, the number of register bits needed to describe thepulse sequence can be drastically reduced by the repeated streaming ofhigh-fidelity subsequences. This strategy leads to compact registersthat are efficient to implement in hardware, and it provides a desirableperiodic suppression of leakage as a side effect, as discussed below.

Consideration of the trade-off between the number of register bits andthe performance of the subsequence would help select the length of thesubsequence. If the length is selected too short, the search space willlikely be too restricted to produce high-fidelity gates, and the numberof controllable qubit frequencies will likely be decreased (see FIG. 4).In some implementations, a good balance may be achieved for subsequencesof approximately 35 to 55 bits, although other values may be possible.The number of subsequence repetitions, and thus the overall length ofthe gate, may be set by the size of the coherent rotation δθ imparted tothe qubit per SFQ pulse. The relationship between gate time and tipangle is T_(g)∝T_(c)/(2δθ), and it is tempting to reduce the gate timeby increasing δθ. However, errors that are second order in δθ willbecome significant for large tip angle. In some simulations, δθ≈0.03 wasfound to be optimal, corresponding to a reasonable coupling capacitancefrom the SFQ driver to the qubit island of order 100 aF for typicaltransmon parameters. For the simulations described here, a Y_(π/2) gatetime of around 12 ns was targeted, although other gate times may bepossible.

With this frequency-matching condition and approach to hardwareoptimization, basic subsequences may be constructed as follows. Given anumber of clock cycles N_(c)′ and qubit cycles N_(q)′, for each clockcycle i∈[0,N′_(c)], an SFQ pulse may be applied on a given clock edgeprovided the pulse induces a rotation in the positive y-direction. Thesubsequence may then be repeated an appropriate number of times toachieve the target rotation. Explicitly, an SFQ pulse is delivered tothe qubit on the k^(th) clock edge of the subsequence provided thefollowing condition is fulfilled:

$\begin{matrix}{{( {N_{q}^{\prime} \cdot \frac{k}{N_{c}^{\prime}}} ){{mod}1}} \leq {{1/4}\mspace{14mu}{or}} \geq {3/4.}} & (22)\end{matrix}$

This class of subsequences is expected to yield reasonably high fidelitybecause it has a palindrome structure, which implies that pulses aredelivered to the qubit as symmetric pairs. For example, the first andlast pulses form the pair (N_(q),0); the second and penultimate pulsesform the pair (N_(q),ω_(q)T_(c)), etc. In general, the sequence containsa pair (N_(q)′,kω_(q)T_(c)) for each k between 0 and 12 that satisfiesEqn. 22.

As an example, a sequence assembled from 10 repetitions of a basicsubsequence using N_(c)′=39 and N_(q)′=8 was simulated. The qubittrajectory is plotted on the Bloch sphere shown in FIG. 4 for qubitsinitialized along the +x (green), +y (purple), and +z (red) directions.The tip angle δθ was selected to achieve a Y_(π/2) rotation in 390 clocksteps. Assuming a qubit anharmonicity of 250 MHz and a 25 GHz SFQ clockfrequency, this sequence achieved fidelity of 99.9% in under 16 ns.Although this scheme for constructing basic subsequences demonstratesthe possibility of controlling multiple qubit frequencies using a singleglobal clock, it is by no means optimal, as the achieved fidelity israther modest. The dominant source of infidelity is leakage from thecomputational subspace. In the following, an approach for suppressingthis leakage is described.

An important aspect of the present SCALLOPS approach is the optimizationalgorithm utilized to eliminate leakage from the computational subspace.Starting with a basic subsequence, as described, bits are flipped tosuppress leakage while preserving the target rotation in the qubitsubspace. The major difficulty in subsequence optimization is that bitflips that reduce leakage may also generally disrupt the rotationachieved in the qubit subspace. This similar to solving a Rubik's cube.That is, when the cube mismatched at the top layer, a naive set ofoperations to complete the top layer will generally disrupt the otherlayers that are already matched. In this case, the difficulty can becircumvented by using operations whose net effect is felt only at thetop layer.

A similar approach may also be used to suppress leakage. Specifically, acorresponding approach is to flip a symmetric pair of bits in thesubsequence and to scale the tip angle δθ to preserve rotation in thequbit subspace. While this latter step might be unexpected given that δθis fixed by the geometric coupling of the SFQ driver to the qubit, itwill be shown that for a given qubit frequency satisfying the matchingcondition Eqn. 21 there exists a high density of high-fidelitysubsequences in the space of tip angles δθ. The strategy will be toallow δθ to vary as a cluster of high-fidelity, low-leakage subsequencesis sought. Then, the subsequences are selected that achieve highestfidelity for the specific value of δθ dictated by the availablehardware.

More formally, this method may be described in terms of a subsequencegraph G=(V, E), where the vertices V represent individual SFQsubsequences with their optimal tip angles δθ and the connections E linksubsequences that are separated by a single symmetric pair of bit flips.Explicitly,

-   -   Each vertex V is described by a subsequence bit pattern S and        its optimal tip angle δθ_(opt)=arg        max_(δ{circumflex over (θ)})F_(avg).    -   Each connection E links subsequences (S, L) that differ by a        single symmetric pair (m,ϕ).

V and E are defined this way with the goal of separating control in thequbit subspace from leakage elimination. That is, navigation through thesubsequence graph G preserves rotation in the qubit subspace, butmovement from vertex to vertex can change leakage out of thecomputational subspace substantially, as seen from Eqn. 18.

For purposes of illustration, a trivial example of this optimizationapproach is shown in FIG. 5. In the figure, each vertex of the graphrepresents a 10-clock cycle (2-qubit cycle) subsequence with a distincttip angle δθ in the qubit subspace (considering a 25 Gz clock and a 5GHz qubit). The shaded regions reflect windows in which positivey-rotations can be induced by the application of SFQ pulses. Verticesthat are connected differ by a single symmetric pair (m,ϕ), which labelsthe connection. For example, the subsequence at the top of the figurediffers from its neighbor on the left by the pair

$( {3,\frac{12\pi}{5}} ),$corresponding to pulses applied on the sixth and ninth clock edgesfollowing initiation of the sequence (i.e. clock edge zero).

With this definition of the subsequence graph G, finding high-fidelitysubsequences begins with a basic subsequence S as defined above. Suchsubsequence serves as the entrance point to the subsequence graph. Then,all vertices adjacent to S are explored, greedily moving to the vertexwith the highest fidelity. This greedy move is then repeated until wereach a local fidelity maximum, which typically takes around 5 to 10steps. In general, 5-8 repetitions of such high-fidelity subsequenceswill yield gates with fidelity greater than 99.99% in a total sequencetime under 12 ns.

The subsequences assembled using the optimization algorithm describedabove are not yet sufficient for experimental implementation because thetip angle per SFQ pulse is allowed to vary during the search. Inpractice, the tip angle is determined by the coupling capacitance of theSFQ driver to the transmon qubit and cannot be exquisitely controlledduring fabrication, or varied in situ following fabrication. To addressthis potential issue, a larger region of the subsequence graph may beexplored to identify a large ensemble of high-fidelity candidatesubsequences corresponding to a range of optimal tip angle δθ_(opt). Foreach of these subsequences, high-fidelity rotations (e.g. withinfidelity under 10⁻⁴) are achieved over a range of δθ, so that it isstraightforward to identify from this ensemble specific subsequencesthat yield high fidelity for a fixed δθ. More specifically, all verticeswith fidelity lower than 99.99% may be ignored, and a standardbreadth-first search may be performed to traverse the remaining verticesof the graph. This leads to a set of characterized subsequences referredto as the subsequence neighborhood.

Referring specifically to FIG. 6, an example plot of infidelitiesachieved versus the tip angle δθ for a subsequence neighborhoodassociated with a 4.65200 GHz qubit is shown. In the plot, eachhorizontal bar represents a unique subsequence. The bars are centeredhorizontally at the optimum tip angle δθ_(opt), and the verticalposition of the bars represents the minimum subsequence infidelity. Thehorizontal extent of each bar denotes the range of δθ over which theinfidelity of the subsequence remains below 10⁻⁴. For each individualsubsequence in the neighborhood, high fidelity is reached for only asmall range of tip angles around the optimal value. However, given afixed value of δθ, numerous subsequences are available that achieve gatefidelity well beyond the target of 99.99%. This is true for SFQ tipangle spanning a broad range from 0.03 to 0.06, which is more thanenough to accommodate any inaccuracy in the design of the couplingcapacitance between the SFQ driver and the transmon.

For purposes of illustration, the above-described neighborhood searchwas performed for 21 different frequencies satisfying the matchingcondition given by Eqn. 21. The results are shown in FIG. 7. In thefigure, each subsequence is labeled below with the frequency of thetarget qubits, and above with the achieved gate fidelity (in units of10⁻⁴) and the number of repetitions required to achieve the Y_(π/2)gate. For each sequence, time flows upward, where red (grey) barscorrespond to clock cycles during which SFQ pulses are applied(omitted). The SCALLOP subsequences span 21 qubit frequencies, and sharea fixed tip angle of 0.032. The frequency spacing of the subsequences isslightly adjusted for improved readability. To note, while a 3-levelmodel of the transmon was used to derive the sequences, the presentedfidelities were calculated for a model incorporating 7 energy levels.

With particular reference to FIG. 8, leakage into the non-computationalstates |2

, |3

, |4

, |5

was examined for the SCALLOP sequence corresponding to the 4.89201 GHzqubit. The sequence involved 6 repetitions of a subsequence consistingof 46 bits. It was observed that the dominant leakage into state |2

was roughly bounded at 10⁻² for initial qubit states spanning thecardinal points on the Bloch sphere. Moreover, as the qubit stateapproached |0

, the leakage into |2

was particularly low, as demonstrated by the curves corresponding toinitial states |z₊

and |x⁻

. While the population of state |2

can approach 10⁻² toward the middle of the sub-sequence, the populationalways drops below 10⁻⁴ at the completion of each subsequencerepetition, as the subsequences are explicitly constructed to minimizeleakage from the qubit subspace. The population of states |3

and |4

was well below 10⁻⁴ throughout, while states |5

and higher had negligible populations.

In addition, the effect of qubit frequency and anharmonicity variationon SCALLOPS gate fidelity was investigated, as shown in FIG. 9. Inparticular, error from frequency drift can be modeled as an ideal gatefollowed by a small precession: U_(fr)(δωT_(g)/ω_(q))Y_(π/2). From Eqn.13, the infidelity of this gate is then approximately (δωT_(g))²/6. Forgate fidelity to degrade by 10⁻⁴, the qubit frequency drift δω/2π mustreach about 300 kHz, given a gate time of 12 ns. This naive estimate isin qualitative agreement with the full simulation results in FIG. 9(upper graph). Note that based on the above argument, microwave-basedqubit gates are expected to display similar sensitivity to qubitfrequency drift. The lower graph of FIG. 9 also shows that SCALLOPS gatefidelity is relatively insensitive to variation in qubit anharmonicity.In a practical system, the anharmonicity of each qubit would becalibrated upon system bring-up. As anharmonicity is set by the transmoncharging energy, it is not expected to fluctuate in time.

The above-described numerical simulations demonstrate coherent qubitcontrol across multiple frequencies using irradiation with classicalbits derived from the SFQ logic family. Using a single global clock at25 GHz to stream pulses from compact registers consisting of 35-55 bits,gate fidelities better than 99.99% were achieved across 21 qubitfrequencies spanning the range from 4.5 to 5.5 GHz. As appreciated fromdescription herein, the present approach provides a number of advantagesand solves a number of problems of prior attempts. For instance, controlsubsequences assembled using methods herein are readily amenable tostorage in compact SFQ-based shift registers. In addition, the presentapproach is an intuitive and efficient method for the derivation ofhigh-fidelity SFQ-based pulse sequences that is readily adapted toarbitrary single-qubit gates. Furthermore, the SCALLOPS method is robustin the sense that large imprecision in the tip angle per SFQ pulse isreadily accommodated by appropriate variation in the subsequencebitstream. Moreover, the control approach is immune to wiring parasiticsignals and offers the possibility for tight integration of alarge-scale quantum array with a proximal classical coprocessor for thepurposes of reducing system footprint, wiring heat load, and controllatency.

Referring now to FIG. 10, a flowchart setting forth steps of a process1000 for controlling qubits, in accordance with the present disclosure,is shown. Steps of the process 1000 may be carried out using anysuitable device, apparatus or system, such as systems described herein.Also, steps of the process 1000 may be implemented as a program,firmware, software, or instructions that may be stored in non-transitorycomputer readable media and executed by a general-purpose, programmablecomputer, processor or other suitable computing device. In someimplementations, steps of the process 1000 may also be hardwired in anapplication-specific computer, processor or dedicated module.

The process 1000 may begin at process block 1002 with assembling a pulsesubsequence using a controller of a quantum computing system.Specifically, the pulse subsequence includes pairs of voltage pulsestimed symmetrically with respect to the period corresponding to thequbit frequency of a qubit. By way of example, the pulse subsequence mayinclude approximately between 30 and 60 classical bits. As described,the assembled pulse subsequence may be optimized to minimize leakagefrom the computational subspace of the qubit.

Then, at process block 1004, a pulse sequence is generated using arepetition of a pulse subsequence, thereby reducing the resourcesrequired. In some aspects, the pulse sequence is configured to perform agate on the qubit, although other control operations may be possible.The pulse sequence may then be applied using the SFQ driver to controlthe qubit, as indicated by process block 1006.

As described, in some aspects, multiple qubits in a qubit architecturemay be controlled using the SFQ driver. To achieve this, the pulsesubsequence assembled at process block 1002 should satisfy the frequencymatching condition in Eqn. 20. That is, the qubit frequency for eachqubit must be tuned such that a gate time associated with the pulsesequence corresponds to both an integer number of SFQ clock cycle and aninteger number of qubit cycle.

The present invention has been described in terms of one or morepreferred embodiments, and it should be appreciated that manyequivalents, alternatives, variations, and modifications, aside fromthose expressly stated, are possible and within the scope of theinvention.

The invention claimed is:
 1. A quantum computing system comprising: aqubit architecture comprising a superconducting qubit described by aqubit frequency; and a single flux quantum (“SFQ”) driver coupled to thesuperconducting qubit, wherein the SFQ driver is configured to provide apulse sequence to control the superconducting qubit, the pulse sequencebeing generated using a repetition of a pulse subsequence that comprisespairs of voltage pulses timed symmetrically with respect to a periodcorresponding to the qubit frequency.
 2. The system of claim 1, whereinthe superconducting qubit comprises a transmon qubit.
 3. The system ofclaim 1, wherein the at least one transmon qubit is capacitively coupledto the SFQ driver.
 4. The system of claim 1, wherein the system furthercomprises a controller configured to direct the SFQ driver to providethe pulse sequence to the superconducting qubit in the qubitarchitecture.
 5. The system of claim 4, wherein the controller isfurther configured to assemble the pulse subsequence using approximatelybetween 30 and 60 classical bits.
 6. The system of claim 4, wherein thecontroller is further configured to assemble the pulse subsequence toperform a gate on the superconducting qubit.
 7. The system of claim 4,wherein the controller is further configured to optimize the pulsesubsequence to minimize leakage from a computational subspace of thesuperconducting qubit.
 8. The system of claim 4, wherein the controlleris further configured to control multiple qubits in the qubitarchitecture using the SFQ driver by assembling the pulse subsequence tosatisfy a frequency matching condition.
 9. A method for controllingsuperconducting qubits, the method comprising: assembling, using acontroller of a quantum computing system, a pulse subsequence thatcomprises pairs of voltage pulses timed symmetrically with respect to aperiod corresponding to a qubit frequency of a superconducting qubit inthe quantum computing system; generating, using the controller, a pulsesequence using a repetition of a pulse subsequence; and controlling thesuperconducting qubit by applying the pulse sequence to thesuperconducting qubit using a single flux quantum (“SFQ”) driver coupledthereto.
 10. The method of claim 9, wherein the method further comprisescontrolling at least one transmon qubit that is capacitively coupled tothe SFQ driver.
 11. The method of claim 9, wherein the method furthercomprises assembling the pulse subsequence using approximately between30 and 60 classical bits.
 12. The method of claim 9, wherein the methodfurther comprises assembling the pulse subsequence to perform a gate onthe superconducting qubit.
 13. The method of claim 9, wherein the methodfurther comprises optimizing the pulse subsequence to minimize leakagefrom a computational subspace of the superconducting qubit.
 14. Themethod of claim 9, wherein the method further comprises controllingmultiple qubits in the qubit architecture of the quantum computingsystem using an SFQ driver coupled thereto.
 15. The method of claim 14,wherein the method further comprises tuning the qubit frequency forselected qubits such that a gate time associated with the pulse sequencecorresponds to both an integer number of SFQ clock cycles and an integernumber of qubit cycles.